deformation of a singularity of type e8 and mordell-weil lattices in characteristic 2

Math. Nachr. 283, No. 7, 1037 – 1053 (2010) / DOI 10.1002/mana.200710057

Deformation of a singularity of type E8 and Mordell-Weil lattices incharacteristic 2

Hiroyuki Ito∗1

1 Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527,Japan

Received 22 March 2007, revised 2 November 2007, accepted 1 January 2008Published online 25 May 2010

Key words Mordell-Weil lattices, ADE singularities, positive characteristic, supersingular K3 surfacesMSC (2000) 14J27, 14B07

In this paper, we study the Mordell-Weil lattices of the family of elliptic surfaces which is arising from the E48

singularity, one of the ADE singularities in characteristic 2. And we construct a subfamily of the universalfamily of supersingular K3 surfaces in characteristic 2 as an application.

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1 Introduction

The theory of Mordell-Weil lattices was independently studied by Shioda and Elkies around 1990, and wasapplied in various fields after that. In his paper [15], Shioda very precisely studied in characteristic 0 Mordell-Weillattices of rational elliptic surfaces in view of the deformation of singularities. After that, Shioda and Usui [16]defined the notion of excellent family to study families of Mordell-Weil lattices and these degenerations.

It is natural to extend what Shioda did in [15] into positive characteristic cases. In doing so, one should takesome peculiar phenomena in low characteristics into account. More precisely, the classification of ADE singular-ities in characteristic 2, 3 and 5, is different from the classification in characteristic 0, and the existence of wildramifications plays a role here. In this paper, we do not give a precise definition of degeneration of Mordell-Weillattices in positive characteristic, instead simply investigate the family of Mordell-Weil lattices arising from thedeformation of the singularity of type E4

8 in characteristic 2. In this case, we give an almost complete descriptionof strata in the deformation family and give some applications. There are five non-isomorphic singularities oftype E8, say E0

8 , E18 , E2

8 , E38 , E4

8 , in characteristic 2 (see Section 3 for their equations). The reason why wechoose E4

8 from these singularities is based on the following observation. By the Kodaira-Neron theory, an ADEsurface-singularity can be naturally associated with an (quasi-)elliptic surface over P1 which has a singular fiberof type corresponding to the singularity. Then, E0

8 , E18 , E2

8 , E38 , E4

8 correspond to a rational quasi-elliptic sur-face, an elliptic surface with κ = 1, an elliptic K3 surface, a rational elliptic surface with constant J-function, arational extremal elliptic surface, respectively.

Here is the plan of this paper. After this introduction, we very briefly review in Section 2 the essential theoremby Shioda on Mordell-Weil lattices. We also recall some definitions and their properties such as conductors andwild ramifications. After reviewing the ADE-singularities in characteristic 2 in Section 3, we precisely studyE4

8 -familyin Sections 4. Finally, we look into three typical subfamilies of E48 -families and give applications for

construction of families of supersingular K3 surfaces.

2 Review of Mordell-Weil lattices and necessary definitions

Let k be an algebraically closed field in characteristic p ≥ 0 and let f : X −→ C be a relatively minimal ellipticsurface over k with a section O. That is, let X (resp. C) be a nonsingular projective surface (resp. curve) over

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1038 Ito: Mordell-Weil lattices

k and that does not have (−1) curves contained in its fibers. Throughout this paper, we assume that an ellipticsurface f : X −→ C has at least one singular fiber.

For an elliptic surface f : X −→ C we define the Mordell-Weil group MW(X/C) to be the groups of sectionsof f . From the Mordell-Weil theorem (proved by Lang and Neron [6] in this case) it follows that MW(X/C) isfinitely generated. We call the rank of MW(X/C) the Mordell-Weil rank.

For the relation between the Mordell-Weil group MW(X/C) and the Neron-Severi group NS(X) of X , thereis the famous isomorphism

MW(X/C) ∼= NS(X)/T,

where T is the trivial sublattice of NS(X) generated by the O-section, a general fiber and irreducible componentsof fibers which do not intersect with the O-section. Note that this isomorphism make a connection between“arithmetic” part and “geometric” part of elliptic surfaces.

On the Mordell-Weil group of an elliptic surface, Shioda defined a nondegenerate positive-definite pairing onMW(X/C)/(tor), and called MW(X/C)/(tor) together with the pairing the Mordell-Weil lattice.

Definition-Theorem 2.1 (Shioda [14]) There exist a unique group homomorphism

ϕ : MW(X/C) −→ T⊥ ⊗ Q ⊂ NS(X) ⊗ Q

with the following properties:

(i) For any P ∈ MW(X/C), ϕ(P ) ≡ (P ) mod TQ,

(ii) Kerϕ = MW (X/C)tor,

where T⊥ is the orthogonal complement of T in the Neron-Severi group NS(X) using the intersection pairingand (P ) is the image of the section P . Let us define a pairing 〈P, Q〉 for each P, Q ∈ MW(X/C) by

〈P, Q〉 := −(ϕ(P ), ϕ(Q)),

where the right-hand side is the intersection pairing extended linearly on NS(X)Q.Then this pairing makes (MW(X/C)/(tor), 〈 , 〉) into a positive-definite lattice and (MW(X/C)◦, 〈 , 〉)

into an even integral, positive-definite (sub)lattice, where MW(X/C)◦ is the subgroup of MW (X/C) whichconsists of the sections (P ) such that (P ) and (O) intersect in each reducible fiber the same component of thatfiber. We call these lattices (MW(X/C)/(tor), 〈 , 〉) and (MW(X/C)◦, 〈 , 〉) the Mordell-Weil lattice andnarrow Mordell-Weil lattice, respectively, of the elliptic surface X/C.

Remark 2.2 One can give ϕ(P ) explicitly by using divisors on X and the intersection matrices of degeneratefibers (see Shioda [14]).

Since we concentrate the case for rational elliptic surfaces in this paper. We give the following beautifulstructure theorem.

Theorem 2.3 (Shioda [14]) Suppose X/P1 is a rational elliptic surface, then the Mordell-Weil latticeMW(X/P1)/(tor) is isomorphic to the dual lattice of T⊥ with opposite inner product and the narrow Mordell-Weil lattice MW(X/P1)◦ is isomorphic to the lattice T⊥ with opposite inner product.

MW(X/P1)/(tor) ∼= (T⊥)∗ ⊂ 〈O, F 〉⊥ ∼= E8

∪ ∪MW(X/P1)◦ ∼= (T⊥)

Moreover, (T⊥)∗ (and T⊥) is a sublattice of 〈O, F 〉⊥, where the orthogonal complement of 〈O, F 〉 takes inthe Neron-Severi group with intersection pairing and 〈O, F 〉⊥ is isomorphic to the lattice of type E8.

Let us call the lattice 〈O, F 〉⊥ as the frame.

Remark 2.4 The torsion part of MW (X/C) can also be written explicitly by using the lattices correspondingto the degenerate fibers as follows (cf. [14])

MW (X/C)tor∼= primitive closure of

⊕v∈R Tv in E8⊕

v∈R Tv,

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Math. Nachr. 283, No. 7 (2010) / www.mn-journal.com 1039

where R is the set of point v of C over which the fiber f−1(v) degenerates and Tv is the lattice corresponding tothe degenerate fiber f−1(v).

Therefore, the Mordell-Weil lattices and the torsion part of the Mordell-Weil groups are completely determinedby the configuration of singular fibers and its embedding into E8 as a lattice.

For more beautiful results in charcteristic 0, see Shioda [14].Next, we are going to introduce an important invariant of a fiber of given elliptic surface, called the conductor,

to treat the geometry of an elliptic surfaces.Let f : X −→ C be an elliptic surface as before. For each v ∈ C, we define the local conductor of the fiber

f−1(v) over v to be one of 0, 1 or 2 + δv according to the case that the fiber is smooth, the fiber is singuar and ofmultiplicative, or the fiber is singular and of additive, where δv is so-called the Swan conductor to measure thewildness of the fiber which we will explain below.

Before explaining the Swan conductor, we define the glocal conductor of the elliptic surface f : X −→ C tobe the sum of local conductors defined as above over every v ∈ C.

Definition 2.5 We call the sum of local conductors as a global conductor, that is,

global conductor :=∑v∈C

(local conductor of f−1(v)

).

For each fiber of the elliptic surfaces, the Swan conductor is defined in terms of the action of the higherramification group of the Galois extension induced from the �-torsion points. (For precise definition, see ChapterIV, Section 10 of Silverman [17].) Instead of giving the precise definition of the Swan conductor, we give thefamous Ogg’s formula to calculate this Swan conductor from a minimal Weierstrass equation of elliptic surfaces.Let us recall this formula from [8].

Proposition 2.6 Let f : X −→ C be an elliptic surface. For every v ∈ C, the local conductor including theSwan conductor of the fiber over v is calculated as follows

ordv(Δ) =(local conductor of f−1(v)

)+ mv − 1,

where Δ is the discriminant divisor of the generic fiber which is an elliptic curve over the function field of C andmv is the number of irreducible components of the fiber f−1(v) over v.

Thus, summing up these orders of the discriminant divisor over all points of C, we get the global version ofOgg’s formula to calculate the global conductor of an elliptic surface.

Proposition 2.7 The global conductor of an elliptic surface is equal to the following invariant of the ellipticsurface: ∑

v∈C

ordv(Δ) −∑v∈C

(mv − 1) = deg(Δ) −∑v∈C

(mv − 1).

Once an elliptic surface is given by the defining equation, then one can easily calculate the discriminant divisorand one easily know what type of singular fibers appear, thus one can calculate every thing in the Ogg’s formulaexcept for the Swan conductor, as a result one gets the Swan conductor itself.

Note that the Swan conductor measures the wildness of ramification of the fiber and it is 0 except when thecharacteristic is 2 or 3 and the fiber is of additive type.

Let us concentrate to the characeristic 2 case. From the precise calculation of the singular fibers of ellipticsurfaces which is given by the defining equation, the Swan conductor δv may be strictly positive, only when thefiber is of type II, III, III∗, II∗ or I∗n with n ≥ 0 (cf. Ogg [8]).

Thus, when the singular fiber over v ∈ C is of type T and the Swan conductor is δv we often use the notationfor this singular fiber as Tδv . For the possible types of singular fibers with Swan conductors in the case of rationalelliptic surfaces, we will give the complete list in the Section 4.

Example 2.8 Let us consider the elliptic surfaces corresponding to the following equation

y2 + tαxy = x3 + t5 (Z � α > 0).

Then the Euler-Poincare characteristic of this surface is just α and the simple calculation shows that the singularfiber over t = 0 (resp. t = ∞) is of type II∗6α−5 (resp. I6α−5).

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Example 2.9 Let us consider the quasi-elliptic surface corresponding to the following equation

y2 = x3 + t5.

Then this surface has the singular fiber of type II∗0 (resp. II0) over t = 0 (resp. t = ∞) (see Ito [4]). For thequasi-elliptic surface, its Mordell-Weil group consists of only p-torsions, thus the δv must be 0 from the definition.

3 ADE singularities in characteristic 2

In this section, we recall the classification of ADE surface-singularities in characteristic 2 and the adjacencyrelations between them from the works by Artin [2] and Greuel-Kroning [3].

Let k be still an algebraically closed field in characteristic 2 and k[[x, y, z]] denote the formal power seriesring. Two power series f, g ∈ k[[x, y, z]] are called contact equivalent if the corresponding local k-algebrask[[x, y, z]]/(f) and k[[x, y, z]]/(g) are isomorphic to each other.

Definition 3.1 A formal power series f is called an ADE surface-singularity (or 2-dimensional simple sin-gularity) in charactersitic 2 if it is contact equivalent to one of the normal forms in Table 1.

Type of singularities Normal form dimension of thesemiuniversal deformation

Ak (k ≥ 1) zk+1 + xy k when k is even,k + 1 when k is odd

D02m (m ≥ 2) z2 + x2y + xym 4m

Dr2m (m ≥ 2, 1 ≤ r ≤ m − 1) z2 + x2y + xym + xym−rz 4m − 2r

D02m+1 (m ≥ 2) z2 + x2y + ymz 4m

Dr2m+1 (m ≥ 2, 1 ≤ r ≤ m − 1) z2 + x2y + ymz + xym−rz 4m − 2r

E06 z2 + x3 + y2z 8

E16 z2 + x3 + y2z + xyz 6

E07 z2 + x3 + xy3 14

E17 z2 + x3 + xy3 + x2yz 12

E27 z2 + x3 + xy3 + y3z 10

E37 z2 + x3 + xy3 + xyz 8

E08 z2 + x3 + y5 16

E18 z2 + x3 + y5 + xy3z 14

E28 z2 + x3 + y5 + xy2z 12

E38 z2 + x3 + y5 + y3z 10

E48 z2 + x3 + y5 + xyz 8

Table 1 Normal forms of ADE surface-singularities in characteristic 2

Remark 3.2 The above normal forms of ADE surface-singularities in characteristic 2 are exactly the normalforms of rational double points which were classified by Artin [2].

Next, we recall deformation relations and adjacencies, between the ADE surface-singularities in character-istic 2. It is known that adjacencies between isolated n-dimensional ADE singularities in characteristic p areunderstood except for the case (n, p) = (2, 2) (cf. [3], [5]).



Although there are some unexpected adjacencies for 2-dimensional case in characteristic 2, the adjacencieswhich appear in our situation are analogous to the characteristic 0 case. Here we give the necessary adjacencydiagram of these singularities which will appear in our family (cf. Corollary 4.5).

A1 A2�� A2 A3

�� A3 A4�� A4 A5

�� A5 A6�� A6 A7

�� A7 A8�� A8 . . .��A3 A4

��A3

D14

��

��

��

� A4

D14

A4

D14

A4 A5��A4

D15

��

��

��

� A5

D15

A5

D15

A5 A6��A5

D26

��

��

��

� A6

D26

A6

D26

A6 A7��A6

D27

��

��

��

� A7

D27

A7

D27

A7 A8��A7

D38

��

��

��

� A8

D38

A8

D38

A8 . . .��A8

. . .

��

��

��

�� . . .

. . .

. . .

. . .D14 D1

5�� D1

5 D26

�� D26 D2

7�� D2

7 D38

�� (D38) . . .��D1

5 D26

��D15

E16

��

��

��

�D2

6

E16

D26

E16

D26 D2

7��D2

6

E37

��

��

��

�D2

7

E37

D27

E37

D27 D3

8��D2

7

E48

��

��

��

�D3

8

E48

D38

E48E1

6 E37

�� E37 E4

8��

Later we will verify that these adjacencies except D38 actually occur and give explicit equations for deforma-

tions in Section 5.

4 E48-family in characteristic 2

Let us look into the family of rational elliptic surfaces given by the deformation of the singularity of type E48 in

characteristic 2. We consider the singularity defined by the affine equation in A3k

y2 + txy = x3 + t5 (4.1)

and the semiuniversal deformation of this singularity with parameter λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8k,

y2 + txy + (p0 + p1t)y = x3 + qx + t5 + r4t4 + r3t

3 + r2t2 + r1t + r0. (4.2)

Then the discriminant of a general member of this deformation is the following

Δλ(t) = t11 + r4t10 + r3t

9 + r2t8 + r1t

7 +(r0 + p1

(q + p2

1

))t6 + p0(q + p2

1)t5

+((

q + p21

)2 + p20p1

)t4 + p3

0t3 + p4

0.(4.3)

For each λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8k, Equation (4.2) defines a possibly singular surface in the affine

space A3k and we can relate this affine surface with the elliptic surface fλ : Xλ −→ P1 unique up to isomorphism.

More precisely, one can think of Equation (4.2) as a defining equation of an elliptic curve Eλ over the functionfield k(t) of one variable, and consider fλ : Xλ −→ P1 as the Kodaira-Neron model of this elliptic curve overk(t). On the other hand, for each elliptic surface fλ : Xλ −→ P1 the surface Xλ−{O}∪f−1

λ (∞) is the minimalresolution of the affine surface with a singularity at the origin defined by the affine equation (4.2) in A3

k, whereO is zero section of fλ and f−1

λ (∞) is the fiber of fλ over ∞ ∈ P1k.

Moreover, we have the following isomorphism under these relations,

MW (Xλ/P1k) ∼= Eλ(k(t))

where Eλ(k(t)) is the Mordell-Weil group of the elliptic curve Eλ over k(t) in the usual sense.

Lemma 4.1 The following conditions are equivalent to each other for any parameter λ ∈ A8k.

(i) The affine surface in A3k defined by the equation (4.2) is smooth.

(ii) The elliptic surface Xλ has no reducible fibers.

(iii) The Mordell-Weil lattice MW (Xλ/P1) is isomorphic to the lattice of type E8.



Let us define the discriminant locus D in the parameter space A8k to be the locus for which the corresponding

affine surface defined by Equation (4.2) is not smooth.For general λ ∈ A8

k, the equation Δλ(t) = 0 has 11 simple roots and hence the singular fibers of Xλ are 12I1’s, thus λ �∈ D. It is clear that the discriminant locus D is contained in the locus where the resultant of Δ(t) andΔ′(t) vanishes. However, note that the locus of vanishing resultant of Δλ(t) and Δ′

λ(t) is strictly bigger than D.The proof of this lemma is very similar to characteristic 0 case (cf. Shioda [15]).Now we have basic results on the degenerate fibers in the E4

8 -family.

Proposition 4.2 Let us write t as a local parameter of P1k. The type of degenerate fibers in the E4

8 -family arethe following:

(i) The singular fiber over t = ∞ is always of type I1, that is, a rational curve with a node.

(ii) The remaining singular fibers are all located over the point on P1k which are roots of the equation

Δλ(t) = 0.(iii) Let us call a root of Δλ(t) = 0 as α and the multiplicity of the root α as να, that is,

να := vt−α(Δλ(t)),

where vt−α stands for the (t − α)-adic valuation in k(t). Then, for each root α �= 0 of Δλ(t) = 0, thesingular fiber over t = α is of type Iνα (1 ≤ να ≤ 9). And the type of the singular fiber over t = 0 isone of the following types when p0 = 0 :

II2, II4, II5, III1, III3, IV0, IV∗0, III∗1, II∗1, I∗0,2, I∗0,3, I∗1,1, I∗2,2, I∗3,2.

P r o o f. Note that the defining equation of Xλ is (4.2), thus we have the defining equation of the fiber overt = ∞ as follows:

Y 2 + XY +(p0τ

3 + p1τ2)Y = X3 + qτ4X + r0τ

6 + r1τ5 + r2τ

4 + r3τ3 + r4τ

2 + τ

where τ is the local parameter at t = ∞, that is, τ = 1t , and we also have the discriminant Δ(τ) as

Δ(τ) = p40τ

12 + p30τ

9 +((

q + p21

)2+ p2

0p1

)τ8 + p0

(q + p2

1

)τ7

+(r0 + p1

(q + p2

1

))τ6 + r1τ

5 + r2τ4 + r3τ

3 + r4τ2 + τ,

thus we get ⎧⎪⎨⎪⎩

vτ

(r0τ

6 + r1τ5 + r2τ

4 + r3τ3 + r4τ

2 + τ)

= 1,

vτ

(Δ(τ)

)= 1,

vτ

(p0τ

3 + p1τ2)

> 0.

Therefore we have (i) by Tate’s algorithm ([18], [8]). (ii) is clear. Since the coefficient of xy in the definingEquation (4.2) is equal to t, the additive fiber can exist only over t = 0 by Tate’s algorithm. And the typesof singular fibers over t �= 0,∞ which are all semistable fibers are easily determined by the multiplicities ofthe discriminant Δλ(t) at the roots using Tate’s algorithm again. Remaining calculations follow from Tate’salgorithm for the type of singular fibers and Ogg’s method for calculating Swan conductors. We only mentionhere the following key formula for a change of coordinates for p0 = 0 case, which will be useful for determiningthe strata.

Lemma 4.3 Equation (4.2) with p0 = 0 can be transfered into

Y 2 + tXY +(p1 + q

12

)tY

= X3 + q14 tX2 +

{q

14

(p1 + q

12

)+ r

120

}tX + t5 + r4t

4 + r3t3 + r2t

2 +{

r1 +(p1 + q

12

)r

120

}t

by the change of coordinates X = x + q12 , Y = y + q

14 x + q

34 + r

124 t2 + r

122 t + r

120 .



Corollary 4.4 Suppose Xλ has no reducible fibers. Then Xλ has the following configuration of singularfibers:

(I1 × 12), (II2, I1 × 8), (II4, I1 × 6), (II5, I1 × 5).

From the list in the above Proposition 4.2(iii), we have the list of singularities which appears as singularities onthe affine surface corresponding to an elliptic surface in E4

8 -family. Thus precise calculations give the followingcorollary.

Corollary 4.5 The singularities appear in the semiuniversal deformation of the singularities of type E48 are

the following:

Al (1 ≤ l ≤ 8), D14 , D1

5, D26, D2

7, E16 , E3

7 , E48 .

One can observe also that all Dynkin diagrams of rank ≤ 7 appear in the list.

Remark 4.6 W. Lang studied the possibility and existence of the configuration of degenerate fibers for rationalelliptic surfaces in characteristic 2 completely in [7]. According to Lang’s list, 34 configurations out of 35 con-figurations of semistable type appear, 54 configurations out of 81 configurations of non-constant non-semistabletype appear in our E4

8 -family.

Another immediate consequence of Proposition 4.2(iii) is the condition for semistability of the fibration.

Corollary 4.7 For every elliptic surface Xλ −→ P1k, all the singular fibers are semistable if and only if

p0 �= 0.

In viewing the defining equation of Xλ precisely, we can find that it does not have rational 2-torsion.

Proposition 4.8 2MW (Xλ/P1) = {0} for every λ ∈ A8k.

P r o o f. Let P = (x(t), y(t)) be k(t)-rational point of fλ : Xλ −→ P1, then we have

−P = (x(t), y(t) + tx(t) + (p0 + p1t))

by the additive formula of elliptic curves.So, P is 2-torsion if and only if P �= O and tx(t) + (p0 + p1t) = 0, that is, x(t) = p1 + p0t

−1. Let ussubstitute x(t) = p1 + p0t

−1 to Equation (4.2) of Xλ, we get

y(t)2 =(p1 + p0t

−1)3 + q

(p1 + p0t

−1)

+ t5 + r4t4 + r3t

3 + r2t2 + r1t + r0,

which is impossible because y(t) should be in k(t).

Since an elliptic surface Xλ has no additive fiber if and only if p0 �= 0, let us write the semistable locus of A8k

as S,

S :={λ ∈ A8

k | p0 �= 0}

.

Clearly this locus is generic in the total space A8k.

Since it is too complicated to consider every member of E48 -family, we pick up the most general ones which

dominates other sub-members when one fix the type of singular fiber over t = 0. We call such a member as abasic member of E4

8 -family as follows.

Definition 4.9 For λ ∈ S, an elliptic surface Xλ is called a basic member if the singular fibers consist of oneIn with some 1 ≤ n ≤ 9 and (12 − n) I1’s. We write this Xλ as X(In).

For λ ∈ A8k − S, an elliptic surface Xλ is called a basic member if the singular fiber over t = 0 is of additive

type T and all the other singular fibers are all I1’s. We write this Xλ as X(T ).Note that a basic member is unique up to the distribution of the singular fibers of type I1 over the simple roots

of Δλ(t) = 0 for λ ∈ A8k − S. On the other hand, a semistable basic member has an ambiguity to locate every

singular fiber other than the one over t = ∞, that is, to distribute one nth multiple root and (11−n) simple rootsfrom the roots of Δλ(t) = 0.



Theorem 4.10 On the parameter space A8k of an E4

8 -family, there are exactly 9 semistable basic membersand 14 non-semistable basic members as below:

(i) Semistable basic members areX(I1), X(I2), X(I3), X(I4), X(I5), X(I6), X(I7), X(I8) and X(I9).

(ii) Non-semistable basic members areX(II2), X(II4), X(II5), X(III1), X(III3), X(IV0), X(I∗0,2), X(I∗0,3), X(I∗1,1), X(I∗2,2), X(I∗3,2),X(IV∗

0), X(III∗1) and X(II∗1).

All members in (ii) correspond to the singular fibers exhibited in Proposition 4.2 (iii).

This is essentially the same as Proposition 4.2. To study various E48 -subfamilies consisting of a given basic

member in the previous theorem, we define strata inside A8k.

Definition 4.11 Let X (T ) be a stratum of A8k satisfying the following condition:

For a general λ ∈ X (T ), the corresponding elliptic surface fλ : Xλ −→ P1 is a basic member X(T ),that is, Xλ for a general λ ∈ X (T ) has only one additive fiber of type T over t = 0 and all otherdegenerate fibers are of type I1.

Let us first describe a stratification inside semistable locus S which is open subset of A8k. The stratification

inside S is described using the distribution of multiplicities of 11 roots of the discriminant equation Δλ(t) = 0.Since S = {p0 �= 0} and Δλ(0) = p4

0, the discriminant equation Δλ(t) = 0 of a general member λ ∈ Shas exactly 11 simple nonzero roots. Let α1, . . . , α11 be roots of the equation Δλ(t) = 0 for λ ∈ S, thereare 11 relations between these αi’s and parameters p0, p1, q, r0, . . . , r4 by (4.3), and these relations are clearlyindependent for general λ ∈ A8

k. Thus S is just X (I1) which is 8-dimensional by the above consideration.For 11 simple roots α1, . . . , α11 corresponding to general λ ∈ X (I1) = S, if one specialize λ to λ1 in order to

make α1 coincide α10, then the discriminant equation Δλ1(t) = 0 has 9 simple roots and one double root, thuscorresponding subset of S which contains λ1 as a general member has dimension 7 and is just X (I2).

Similarly, suppose the discriminant equation Δλ(t) = 0 has 11 − l simple roots and one l-tuple root, then thecorresponding Xλ is a general member of X (Il) which has dimension 9 − l, and it has codimension one stratumX (Il+1) whose general member Xλ1 is obtained from the above 11 − l simple roots and one l-tuple root byspecializing one simple root to the unique multiple root.

Thus we have the following theorem.

Theorem 4.12 Semistable locus S of A8k has the stratification as follows

A8k ⊃ S = X (I1) ⊃ X (I2) ⊃ · · · ⊃ X (I8) ⊃ X (I9).

Each stratum X (Il) (l > 1) has codimension 1 inside X (Il−1), thus a stratum X (Il) has dimension 9 − l.

Proof is a monotonous calculation and left to the reader.

Remark 4.13 X (I2) coincides with the locus where the resultant of Δλ(t) and Δ′λ(t) vanishes.

X (I2) ={λ ∈ A8

k | Resultant of Δλ(t) and Δ′λ(t) is equal to zero.

}As a corollary, we find “degeneration” of Mordell-Weil lattices corresponding to the semistable strata.

Corollary 4.14 The Mordell-Weil lattices of semistable basic members X(I9−l) (0 ≤ l ≤ 8) are as in Table 2.

Because we know the exact type of singular fibers of each basic member, a proof simply follows from theresults by Oguiso-Shioda [9]. Note that we have adjacency diagrams between singularities of type Al (1 ≤ l ≤ 8)corresponding to these semistable fibers.



Type of the semistable fiber the Mordell-Weil lattice its narrow Mordell-Weil lattice

I1 × 12 E8 E8

I2, I1 × 10 E∗7 E7

I3, I1 × 9 E∗6 E6

I4, I1 × 8 D∗5 D5

I5, I1 × 7 A∗4 A4

I6, I1 × 6 A∗2 ⊕ A∗

1 A2 ⊕ A1

I7, I1 × 517

⎛⎜⎝2 1

1 4

⎞⎟⎠

⎛⎜⎝ 4 −2

−1 2

⎞⎟⎠

I8, I1 × 4⟨

18

⟩〈8〉

I9, I1 × 3 Z/3Z {0}Table 2 The Mordell-Weil lattices of semistable basic members

Next we describe a stratification on non-semistable locus of A8k by strata corresponding to these basic members

using hypersurfaces of A8k defined below.

P0 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | p0 = 0}

,

P1 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | q + r2 = 0}

,

Q :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | q + p21 = 0

},

R0 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | r0 + q12 (p2

1 + q) = 0}

,

R1 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | r1 + r120 (p1 + q

12 ) = 0

},

R2 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | r2 = 0}

,

R3 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | r3 + q14 r

122 = 0

},

R4 :={λ = (p0, p1, q, r4, r3, r2, r1, r0) ∈ A8

k | r4 = 0}

.

Then by precise analysis of defining equations of these basic members using Tate’s algorithm, we have thestratification on this non-semistable locus and the defining equations for generic members of these strata.

Theorem 4.15 Inside the non-semistable locus P0 = A8k \ S, each stratum and its description using hyper-

surfaces defined above are given in Table 3.Furthermore, these stratum are all affine spaces and the defining equations of the basic members of them are

as in Table 4.

P r o o f. It is enough to check how descend from the stratum X (II2) to each stratum. This is simple calculationusing Tate’s algorithm and the key Lemma 4.3 on change of coordinates in a couple of times.

As a corollary of this stratification, we know the structure of the Mordell-Weil lattice of a general member ofeach strata followed by the work of Oguiso-Shioda [9].

Corollary 4.16 The Mordell-Weil lattice of a general member X(T ) of each strata X (T ) inside the non-semistable locus is as in Table 5.



Stratum its description generic parameter

X (II2) = A8k \ S = A8

k ∩ P0 (0, p1, q, r4, r3, r2, r1, r0)

X (II4) = X (II2) ∩ Q = A8k ∩ P0 ∩ Q

(0, p1, p

21, r4, r3, r2, r1, r0

)X (II5) = X (II4) ∩ R0 = A8

k ∩ P0 ∩ Q ∩ R0

(0, p1, p

21, r4, r3, r2, r1, 0

)X (III1) = X (II2) ∩ R1 = A8

k ∩ P0 ∩ R1

(0, p1, q, r4, r3, r2, r

120

(p1 + q

12

), r0

)X (III3) = X (III1) ∩ Q = X (II4) ∩ R1

(0, p1, p

21, r4, r3, r2, 0, r0

)= A8

k ∩ P0 ∩ R1 ∩ Q

X (IV0) = X (III1) ∩ R0 = A8k ∩ P0 ∩ R1 ∩ R0

(0, p1, q, r4, r3, r2, q

14

(p21 + q

), q

12

(p21 + q

))X (I∗0,2) = X (IV0) ∩ Q = X (III3) ∩ R0

(0, p1, p

21, r4, r3, r2, 0, 0

)= X (II5) ∩ R1 = A8

k ∩ P0 ∩ R0 ∩ R1 ∩ Q

X (I∗0,3) = X (I∗0,2) ∩ R2 = A8k ∩ P0 ∩ Q ∩ R0 ∩ R1 ∩ R2

(0, p1, p

21, r4, r3, 0, 0, 0

)X (I∗1,1) = X (I∗0,2) ∩ R3 = A8

k ∩ P0 ∩ Q ∩ R0 ∩ R1 ∩ R3

(0, p1, p

21, r4, p

121 r

122 , r2, 0, 0

)X (I∗2,2) = X (I∗1,1) ∩ R2 = X (I∗0,3) ∩ R3

(0, p1, p

21, r4, 0, 0, 0, 0

)= A8

k ∩ P0 ∩ Q ∩ R0 ∩ R1 ∩ R2 ∩ R3

X (I∗3,2) = X (I∗2,2) ∩ R4

(0, p1, p

21, 0, 0, 0, 0, 0

)= A8

k ∩ P0 ∩ Q ∩ R0 ∩ R1 ∩ R2 ∩ R3 ∩ R4

X (IV∗0) = X (I∗1,1) ∩ P1

(0, r

122 , r2, r4, r

342 , r2, 0, 0

)= A8

k ∩ P0 ∩ P1 ∩ Q ∩ R0 ∩ R1 ∩ R3

X (III∗1) = X (IV∗0) ∩ R2 = X (I∗2,2) ∩ P1 (0, 0, 0, r4, 0, 0, 0, 0)

= A8k ∩ P0 ∩ P1 ∩ Q ∩ R0 ∩ R1 ∩ R2 ∩ R3

X (II∗1) = X (III∗1) ∩ R4 = X (I∗3,2) ∩ P1 (0, 0, 0, 0, 0, 0, 0, 0)

Table 3 Stratum inside the non-semistable locus

Corollary 4.17 We have the corresponding adjacency diagram on singularities of type A1, A2, D14 , D1

5 , D26,

D27 , E1

6 , E37 , E4

8 as at the end of Section 3.

We also have the corresponding “degenerations of Mordell-Weil lattices” inside the non-semistable locusA8

k \ S which will explain in the next section precisely.

5 Three examples and applications

In this section, we pick up three subfamilies of E48 -family more precisely.

5.1 Semistable family

First of all, let us consider the semistable subfamily S. A general member of S has no reducible fiber as inSection 4, thus it has 12 singular fibers of type I1 and its Mordell-Weil lattice is isomorphic to E8.



Stratum defining equation of the basic member

X (II2) ∼= A7 y2 + txy + p1ty = x3 + qx + t5 + r4t4 + r3t

3 + r2t2 + r1t + r0

X (II4) ∼= A6 y2 + txy + p1ty = x3 + p21x + t5 + r4t

4 + r3t3 + r2t

2 + r1t + r0

X (II5) ∼= A5 y2 + txy + p1ty = x3 + p21x + t5 + r4t

4 + r3t3 + r2t

2 + r1t

X (III1) ∼= A6 y2 + txy + p1ty = x3 + qx + t5 + r4t4 + r3t

3 + r2t2 + r

120 (p1 + q

12 )t + r0

X (III3) ∼= A5 y2 + txy + p1ty = x3 + p21x + t5 + r4t

4 + r3t3 + r2t

2 + r0

X (IV0) ∼= A5 y2 + txy + p1ty = x3 + qx + t5 + r4t4 + r3t

3 + r2t2 + q

14 (p2

1 + q)t + q12 (p2

1 + q)

X (I∗0,2) ∼= A4 y2 + txy + p1ty = x3 + p21x + t5 + r4t

4 + r3t3 + r2t

2


4 + r3t3


4 + p121 r

122 t3 + r2t

2


4

X (I∗3,2) ∼= A1 y2 + txy + p1ty = x3 + p21x + t5

X (IV∗0) ∼= A2 y2 + txy + r

122 ty = x3 + r2x + t5 + r4t

4 + r342 t3 + r2t

2

X (III∗1) ∼= A1 y2 + txy = x3 + t5 + r4t4

X (II∗1) ∼= {point} y2 + txy = x3 + t5

Table 4 Defining equations of basic members of non-semistable stratum

As we saw in Theorem 4.12 and Remark 4.13, we can cut S = X (I1) by the hypersurface defined by theresultant of Δλ(t) and Δ′

λ(t) to get X (I2), and continue to cut X (Ik) by a subvariety of A8k to get X (Ik+1)

which is codimension 1 subvariety of its precedent. Finally we get the stratification on semistable subfamily ofE4

8 -family:

A8k ⊃ S = X (I1) ⊃ X (I2) ⊃ · · · ⊃ X (I8) ⊃ X (I9).

By the explicit calculation, we can determine the Weierstrass equations of the basic members of low dimen-sional loci of this stratification.

Proposition 5.1 (1) The Weierstrass equation of X(I9) is given by

y2 + txy + y = x3 + t5 + t2.

Thus the singular fiber of type I9 is located over t = 1 and I1’s are over t = ω, ω2,∞, where ω is a cubicroot of unity.

(2) The Weierstrass equation of X(I8) is given by

y2 + txy + αy = x3 + t5 + αt2

where α ∈ k \ {0, 1}. Thus the singular fiber of type I8 is located over t = α38 and I1’s are over

t = α13 , α

13 ω, α

13 ω2,∞.



Type of the number of the Mordell-Weil lattice its narrow

the non-semistable fiber fibers of type I1 Mordell-Weil lattice

II2 8 E8 E8

II4 6 E8 E8

II5 5 E8 E8

III1 8 E∗7 E7

III3 6 E∗7 E7

IV0 8 E∗6 E6

I∗0,2 4 D∗4 D4

I∗1,1 4 A∗3 A3

IV∗0 4 A∗

2 A2

I∗0,3 3 D∗4 D4

I∗2,2 2 A∗1⊕2 A⊕2

1

III∗1 2 A∗1 A1

I∗3,2 1 〈14 〉〈4〉

II∗1 1 {0} {0}Table 5 The Mordell-Weil lattices of non-semistable basic members

P r o o f. Suppose the reducible fiber Il (l = 8 or 9) is located over t = α and other irreducible fibers I1 arelocated over t = β1, β2, . . . , β11−l,∞. Since the discriminant Δ(t) must be (t + α)9(t + β1)(t + β2) for X(I9)(resp. (t + α)8(t + β1)(t + β2)(t + β3) for X(I8)) which is equal to (4.3), one can easily calculate to deduce theresult. If one specialize the α to 1, then one gets X(I9).

Note that X (I9) consists of one point by this proposition and X (I8) is isomorphic to A1k. Note also that if one

specialize α to 1 in X (I8) one get X (II1) which consists of one point.Since every member of S corresponds to an elliptic surface over P1, we can take Frobenius base change and

resolution simultaneously to get a new stratification:

A8k ⊃ S = X (I1) ⊃ X (I2) ⊃ · · · ⊃ X (I8) ⊃ X (I9).

More precisely, each X (Il) is the purely inseparable cover of X (Il) of degree 2, thus isomorphic to the affinespace. Note that this purely inseparable cover needs for the simultaneous resolution because all the singularities

appearing on the Frobenius base changed elliptic surfaces are A1 singularities. Therefore, for each μ ∈ X (Il),the corresponding elliptic surface is the resolution of the fiber product fλ : Xλ×P1 P1 −→ P1 of fλ : Xλ −→ P1

and the purely inseparable map P1 −→ P1 of degree 2 with μ = (μ1, μ2, . . . , μ8) and λ = (λ1, λ2, . . . , λ8) =(μ2

1, μ22, . . . , μ

28).

Now we can state the results on this new stratification.

Theorem 5.2 For 1 ≤ l ≤ 9, a general member X(Il) of X (Il) is a supersingular elliptic K3 surface whosesingular fibers consist of one I2l and (12 − l) I2’s. Moreover, its Artin invariant is 10 − l and the Mordell-Weillattice is as in Table 6, where L(2) stands for the lattice L whose pairing is twice of the original pairing on L.



l Type of the Mordell-Weil lattice its narrow Mordell-Weil lattice

the semistable singular fibers

1 I2 × 12 E8(2) ⊕ Z/2Z E8(2)

2 I4, I2 × 10 E∗7 (2) ⊕ Z/2Z E7(2)

3 I6, I2 × 9 E∗6 (2) ⊕ Z/2Z E6(2)

4 I8, I2 × 8 D∗5(2) ⊕ Z/2Z D5(2)

5 I10, I2 × 7 A∗4(2) ⊕ Z/2Z A4(2)

6 I12, I2 × 6 A∗2(2) ⊕ A∗

1(2) ⊕ Z/2Z A2(2) ⊕ A1(2)

7 I14, I2 × 527

⎛⎜⎝2 1

1 4

⎞⎟⎠ ⊕ Z/2Z

⎛⎜⎝ 8 −2

−2 4

⎞⎟⎠

8 I16, I2 × 4⟨

14

⟩⊕ Z/2Z 〈16〉

9 I18, I2 × 3 Z/3Z ⊕ Z/2Z {0}Table 6 The Mordell-Weil lattices of base changed semistable members

P r o o f. By using Ohhira’s machinery [10] on how the type of a singular fiber changes under a purely insep-

arable base change, we can determine the type of singular fibers for each surface X(Il) as in Table 6. Since thefollowing relationship among the Mordell-Weil lattices holds:

MW(X(Il))◦(2) � � ��

MW(X(Il))(2)

MW(X(Il))◦(2)

MW(X(Il))(2)

MW(X(Il))◦(2)

MW(X(Il))(2) � � ��

MW(X(Il))◦(2)

MW(X(Il))(2)

��

��

MW(X(Il))◦

MW(X(Il))

��

��

,

we have the inequality

rk MW(X(Il)) ≤ rk MW(X(Il)).

By Shioda-Tate formula, the left-hand side is equal to 10−2−(l−1) = 9−l and the right-hand side is equal to

ρ(X(Il))−2−{(2l−1)+(12−l)} = ρ(X(Il))−l−13 which is less than or equal to 22−l−13 = 9−l since X(Ik)

is a K3 surface, therefore, the inequalities above are all equalities. We have rk MW(X(Il)) = rk MW(X(Il))

and ρ(X(Ik)) = 22, that is, X(Il) is a supersingular K3 surface. Moreover, the inclusions in the commutativediagram above have all finite indices. We also have MW(X(Il))(2)free = MW(X(Il))free(2) has a finite index

inside MW(X(Il))free. Let us write this index as μ.Then we have

μ2 =detMW(X(Il))free(2)

detMW(X(Il))free

.

When the rank of MW(X(Il)) is positive, that is, 1 ≤ l ≤ 8, the numerator of the right-hand side is equal to29−ll−1 since the rank of MW(X(Il))free is equal to 9 − l and MW(X(Il)) is isomorphic to the dual lattice ofthe orthogonal complement of Al−1 inside E8. The denominator of the right-hand side can be calculated usingthe following lemma.



Lemma 5.3 Let X be an elliptic surface. Then we have the following equality

detMW(X)free =detNS(X)

detT· |MW(X)tor|2,

where T stands for the trivial lattice of X .

P r o o f. From the general theory of Mordell-Weil lattices, the orthogonal complement T⊥ of T in NS(X) isisomorphic to the narrow Mordell-Weil lattice MW(X)◦, thus T + T⊥ has finite index inside NS(X) and thisindex is equal to the index of MW(X)◦ inside MW(X). (Cf. [14].)

Thus by usual linear algebra argument, we have

detT · detT⊥

detNS(X)=

[NS(X) : T + T⊥]2

= |MW(X)/ MW(X)◦|2

= |MW(X)tor|2 det MW(X)◦

detMW(X)free

= |MW(X)tor|2 · detT⊥

detMW(X)free.

By this lemma,

detMW(X(Il))free =−22σ0

−212−l2l|MW(X(Il))tor|2 (5.1)

where σ0 is the Artin invariant of the supersingular K3 surface X(Ik). Summing them up, we get∣∣∣MW(X(Il))tor

∣∣∣ · μ = 211−l−σ0 .

Since the prime-to-p part of a torsion subgroup is not changed by a purely inseparable base extension, we onlyneed to check the existence of 2-torsion and 4-torsion for X(Il).

Lemma 5.4

MW(X(Il)/P1) ⊃ Z/2Z, MW(X(Il)/P1) �⊃ Z/4Z.

P r o o f. Since we have a nontrivial 2-torsion,⎧⎨⎩

x(t) = p1 + p0t2,

y(t) =(p

121 + p

120 t−1

)3

+ q12

(p

121 + p

120 t−1

)+ t5 + r

124 t4 + r

123 t3 + r

122 t2 + r

121 t + r

120 ,

and elementary calculation shows non-existence of a 4-torsion.

Thus we have 2l+σ0 · μ = 210 (1 ≤ l ≤ 8, 1 ≤ σ0 ≤ 9), and σ0 ≤ 10 − l.On the other hand, any two general members of X (Il) are not isomorphic to each other, therefore the moduli

map from X (Il) which is an affine space of 9 − l dimension to the moduli space of supersingular K3 surfaces is

generically injective for each l. Thus the σ0 of a general member of X (Il) is equal to 10− l by the general theoryon the moduli space of supersingular K3 surfaces (cf. [1], [11]).

For l = 9 case, all Mordell-Weil groups appeared here are finite, direct calculation from (5.1) gives σ0 = 1 =10 − 9. Finally, we get σ0 = 10 − l and μ = 1.

Remark 5.5 In [11], Rudakov and Shafarevich constructed the universal family of supersingular K3 surfacesin characteristic 2 by using quasi-elliptic surfaces. Our family realizes a part of their family by using semistableelliptic surfaces.



5.2 Non-semistable family

We treat the subfamily defined by the equation p0 = 0 inside the parameter space A8k. That is, the generic

equation of this subfamily is given by

y2 + txy + p1ty = x3 + qx + t5 + r4t4 + r3t

3 + r2t2 + r1t + r0,

where the 7-dimensional parameter is (p1, q, r4, r3, r2, r1, r0) ∈ A7k.

Since the strata inside this family are rather complicated, we divide this family into two parts, II2-family, the“upper part” of our subfamily, and I∗0,2-family, the “lower part” of our subfamily.

Let us recall the notations P0, P1, Q, R0, R1, R2, R3, R4 as in Section 4 which stand for some hypersurfacesin A8

k. As we showed in Table 3, X (II4) is obtained from X (II2) cutting by Q, X (II5) is obtained from X (II4)cutting by P0, and so on.

Let us draw the picture of these relations inside II2-family with corresponding Mordell-Weil lattices:

X (II4) X (III3)∩R1

��

X (II2)

X (II4)

∩Q

��

X (II2) X (III1)∩R1�� X (III1)

X (III3)

∩Q

��X (II4) X (III3)X (II4) X (III3)X (II4)

X (II5)

∩R0

��

X (III3)

X (II5)

X (III3)

X (II5)

X (III1) X (IV0)∩R0��X (III1)

X (III3)

X (III1)

X (III3)

X (IV0)

X (III3)

X (IV0)

X (III3)

X (I∗0,2)X (I∗0,2)

X (III3)X (III3)X (III3)

X (I∗0,2)

∩R0

��

��

��

X (II5) X (I∗0,2)∩R1 ��

X (IV0)

X (I∗0,2)

∩Q

��

E8 E∗7

��

E8

E8

E8 E∗7

�� E∗7

E∗7E8 E∗7E8 E∗7E8

E8

E∗7

E8

E∗7

E8

E∗7 E∗

6��E∗

7

E∗7

E∗7

E∗7

E∗6

E∗7

E∗6

E∗7

D∗4D∗4

E∗7E∗7E∗7

D∗4

��

��

��

E8 D∗4

��

E∗6

D∗4

��

As in the semistable case, one can consider the Frobenius base extension of this family to get a new familyof supersingular elliptic K3 surfaces except the ones obtained from X (I∗0,2). The elliptic surface obtained by thepurely inseparable base extension of the basic member X(I∗0,2) is a rational elliptic surface with singularities, andits resolution gives an elliptic surface whose configuration of the degenerate fibers is II2, I2 × 4.

Here are the picture with the configurations of singular fibers and the singularities which correspond to theadditive fibers.

(I∗2,4, 6I′2s) (I∗3,3, 6I′2s)��

(I∗0,2, 8I′2s)

(I∗2,4, 6I′2s)��

(I∗0,2, 8I′2s) (I∗1,1, 8I′2s)�� (I∗1,1, 8I′2s)

(I∗3,3, 6I′2s)��

(I∗2,4, 6I′2s) (I∗3,3, 6I′2s)(I∗2,4, 6I′2s) (I∗3,3, 6I′2s)(I∗2,4, 6I′2s)

(III∗5, 5I′2s)��

(I∗3,3, 6I′2s)

(III∗5, 5I′2s)

(I∗3,3, 6I′2s)

(III∗5, 5I′2s)

(I∗1,1, 8I′2s) (IV∗0, 8I′2s)��(I∗1,1, 8I′2s)

(I∗3,3, 6I′2s)

(I∗1,1, 8I′2s)

(I∗3,3, 6I′2s)

(IV∗0, 8I′2s)

(I∗3,3, 6I′2s)

(IV∗0, 8I′2s)

(I∗3,3, 6I′2s)

(II2, 4I′2s)(II2, 4I′2s)

(I∗3,3, 6I′2s)(I∗3,3, 6I′2s)(I∗3,3, 6I′2s)

(II2, 4I′2s)��

��

��

��

(III∗5, 5I′2s) (II2, 4I′2s)��

(IV∗0, 8I′2s)

(II2, 4I′2s)��

D06 D0

7��

D04

D06

��

D04 D0

5�� D0

5

D07

��D0

6 D07D0

6 D07D0

6

E07

��

D07

E07

D07

E07

D05 E0

6��D0

5

D07

D05

D07

E06

D07

E06

D07

Next we look into the subfamily, I∗0,2-family, which is the “lower part” of non-semistable family. This givesa 4-dimensional stratification. Generically, a member of this family can be written as the following Weierstrassequation after a change of parameters:

y2 + txy = x3 + p121 tx2 + r

122 t2x + t5 + r4t

4 + r3t3.



A generic member of this family has a singular fiber of type I∗0,2 over t = 0 and four other singular fibers oftype I1. As before, we describe the specializations inside this family and the corresponding Mordell-Weil lattices:

X (I∗0,3) X (I∗2,2)∩R3

��

X (I∗0,2)

X (I∗0,3)

∩R2

��

X (I∗0,2) X (I∗1,1)∩R3�� X (I∗1,1)

X (I∗2,2)��

X (I∗2,2) X (III∗1)∩P1

��

X (I∗1,1)

X (I∗2,2)

∩R2

��

X (I∗1,1) X (IV∗0)

∩P1�� X (IV∗0)

X (III∗1)

∩R2

��

X (I∗3,2) X (II∗1)∩P1

��

X (I∗2,2)

X (I∗3,2)

∩R4

��

X (I∗2,2) X (III∗1)�� X (III∗1)

X (II∗1)

∩R4

��

D∗4 A∗

1⊕2��

D∗4

D∗4

D∗4 A∗

3�� A∗

3

A∗1⊕2��

A∗1⊕2 A∗

1��

A∗3

A∗1⊕2��

A∗3 A∗

2�� A∗

2

A∗1

��

〈14 〉 {0}��

A∗1⊕2

〈14 〉��

A∗1⊕2 A∗

1�� A∗

1

{0}��

Recall that the dimension of the upper left family X (I∗0,2) is 4 and it decreases when it is specialized by ahypersurface as a direction of each arrow. Thus the family of most right below X (II∗1) has dimension 0. Aftertaking the resolution of the purely inseparable base extension of degree 2, we get a 4-dimensional family ofrational elliptic surfaces. Here are the configurations of degenerate fibers of these basic members and the freepart of Mordell-Weil lattices. All of them have same torsion group Z/2Z as torsion part of Mordell-Weil groups.

(III3, 3I′2s) (I∗0,2, 2I′2s)��

(II2, 4I′2s)

(III3, 3I′2s)��

(II2, 4I′2s) (III1, 4I′2s)�� (III1, 4I′2s)

(I∗0,2, 2I′2s)��

(I∗0,2, 2I′2s) (I∗1,1, 2I′2s)��

(III1, 4I′2s)

(I∗0,2, 2I′2s)��

(III1, 4I′2s) (IV0, 4I′2s)�� (IV0, 4I′2s)

(I∗1,1, 2I′2s)��

(I∗2,2, I2) (III∗1, I2)��

(I∗0,2, 2I′2s)

(I∗2,2, I2)��

(I∗0,2, 2I′2s) (I∗1,1, 2I′2s)�� (I∗1,1, 2I′2s)

(III∗1, I2)��

D∗4 A∗

1⊕2��

D∗4

D∗4

��

D∗4 A∗

1⊕3�� A∗

1⊕3

A∗1⊕2��

A∗1⊕2 〈1

4 〉��

A∗1⊕3

A∗1⊕2��

A∗1⊕3 1

6

(2 11 2

)�� 16

(2 11 2

)

〈14 〉��

A∗1 {0}��

A∗1⊕2

A∗1

��

A∗1⊕2 〈1

4 〉�� 〈14 〉

{0}��

We note that these Mordell-Weil lattices have sublattices, which naturally come from the Mordell-Weil latticesof the X(T )’s with indices μ, which can be calculated similarly in the semistable case. Since these calculationsare simple, we omit them.

For applications of this non-semistable family, we will back on another occasion.

Acknowledgements The author would like to thank Professors M. Hirokado, T. Kajiwara, N. Saito, and A. Schweizer forstimulating discussions on various subjects in positive characteristic including the arguments in this paper. The author thanksthe Mittag-Leffler Institute for the kind hospitality where the part of this work was done. The author also thanks the refereefor many useful comments improving the paper.

The research of the author was partially supported by Grant-in-Aid from Inamori Foundation, Grant-in-Aid for ScientificResearch 17540027, Ministry of Education, Science and Culture, and Grant-in-Aid from Fujii Foundation.

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deformation of a singularity of type e8 and mordell-weil lattices in characteristic 2

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